Simplification d'écritures - Corrigé

Énoncé

Simplifier les écritures suivantes.

$z_1=3{\text{e}}^{-\frac{i\pi }{3}}\times 8{\text{e}}^{\frac{i\pi }{4}}$

$z_2={\displaystyle \frac{-2{\text{e}}^{i\frac{\pi }{2}}}{{\text{e}}^{\frac{i\pi }{7}}}}$

$z_3={\left(2{\text{e}}^{\frac{i\pi }{4}}\right)}^6$

$z_4={\displaystyle \frac{{\left({\text{e}}^{i\pi }\right)}^5}{{\left(4{\text{e}}^{\frac{i\pi }{9}}\right)}^2}}$

Solution

On a : ${\displaystyle \begin{array}{rl}{z}_1& =3{\text{e}}^{-\frac{i\pi }{3}}\times 8{\text{e}}^{\frac{i\pi }{4}}=24{\text{e}}^{-\frac{i\pi }{3}+\frac{i\pi }{4}}=24{\text{e}}^{i(-\frac{\pi }{3}+\frac{\pi }{4})}=24{\text{e}}^{-\frac{i\pi }{12}}.\end{array}}$

On a : $\begin{array}{rl}{z}_2& ={\displaystyle \frac{-2{\text{e}}^{i\frac{\pi }{2}}}{{\text{e}}^{\frac{i\pi }{7}}}}=-2{\text{e}}^{i\frac{\pi }{2}-\frac{i\pi }{7}}=-2{\text{e}}^{i(\frac{\pi }{2}-\frac{\pi }{7})}=-2{\text{e}}^{\frac{5i\pi }{14}}.\end{array}$

On a : $\begin{array}{rl}{z}_3& ={\left(2{\text{e}}^{\frac{i\pi }{4}}\right)}^6=2^6{\text{e}}^{\frac{6i\pi }{4}}=64{\text{e}}^{\frac{3i\pi }{2}}.\end{array}$

On a : $\begin{array}{rl}{z}_4& ={\displaystyle \frac{{\left({\text{e}}^{i\pi }\right)}^5}{{\left(4{\text{e}}^{\frac{i\pi }{9}}\right)}^2}}=\frac{{\text{e}}^{5i\pi }}{4^2{\text{e}}^{\frac{2i\pi }{9}}}=\frac{1}{16}{\text{e}}^{5i\pi -\frac{2i\pi }{9}}=\frac{1}{16}{\text{e}}^{i(5\pi -\frac{2\pi }{9})}=\frac{1}{16}{\text{e}}^{\frac{43i\pi }{9}}=\frac{1}{16}{\text{e}}^{\frac{7i\pi }{9}}.\end{array}$